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Protection of a network by complete secure domination

Year 2022, , 47 - 55, 15.01.2022
https://doi.org/10.13069/jacodesmath.1056581

Abstract

A complete secure dominating set of a graph $G$ is a dominating set $D \subseteq V(G)$ with the property that for each $v \in D$, there exists $F=\lbrace v_{j} \vert v_{j} \in N(v) \cap (V(G)-D)\rbrace$, such that for each $v_{j} \in F$, $( D-\lbrace v \rbrace) \cup \lbrace v_{j} \rbrace$ is a dominating set. The minimum cardinality of any complete secure dominating set is called the complete secure domination number of $G$ and is denoted by $\gamma_{csd}(G)$. In this paper, the bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of $G$ are found and also the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of $G$.

References

  • [1] V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).
  • [2] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).
  • [3] N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).
  • [4] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.
  • [5] J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.
  • [6] E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.
  • [7] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).
Year 2022, , 47 - 55, 15.01.2022
https://doi.org/10.13069/jacodesmath.1056581

Abstract

References

  • [1] V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).
  • [2] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).
  • [3] N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).
  • [4] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.
  • [5] J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.
  • [6] E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.
  • [7] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).
There are 7 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Girish V. Rajasekharaiah This is me 0000-0002-0036-6542

Usha P. Murthy This is me 0000-0001-9855-1887

Umesh Subramanya This is me

Publication Date January 15, 2022
Published in Issue Year 2022

Cite

APA Rajasekharaiah, G. V., Murthy, U. P., & Subramanya, U. (n.d.). Protection of a network by complete secure domination. Journal of Algebra Combinatorics Discrete Structures and Applications, 9(1), 47-55. https://doi.org/10.13069/jacodesmath.1056581
AMA Rajasekharaiah GV, Murthy UP, Subramanya U. Protection of a network by complete secure domination. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):47-55. doi:10.13069/jacodesmath.1056581
Chicago Rajasekharaiah, Girish V., Usha P. Murthy, and Umesh Subramanya. “Protection of a Network by Complete Secure Domination”. Journal of Algebra Combinatorics Discrete Structures and Applications 9, no. 1 n.d.: 47-55. https://doi.org/10.13069/jacodesmath.1056581.
EndNote Rajasekharaiah GV, Murthy UP, Subramanya U Protection of a network by complete secure domination. Journal of Algebra Combinatorics Discrete Structures and Applications 9 1 47–55.
IEEE G. V. Rajasekharaiah, U. P. Murthy, and U. Subramanya, “Protection of a network by complete secure domination”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 47–55, doi: 10.13069/jacodesmath.1056581.
ISNAD Rajasekharaiah, Girish V. et al. “Protection of a Network by Complete Secure Domination”. Journal of Algebra Combinatorics Discrete Structures and Applications 9/1 (n.d.), 47-55. https://doi.org/10.13069/jacodesmath.1056581.
JAMA Rajasekharaiah GV, Murthy UP, Subramanya U. Protection of a network by complete secure domination. Journal of Algebra Combinatorics Discrete Structures and Applications.;9:47–55.
MLA Rajasekharaiah, Girish V. et al. “Protection of a Network by Complete Secure Domination”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 9, no. 1, pp. 47-55, doi:10.13069/jacodesmath.1056581.
Vancouver Rajasekharaiah GV, Murthy UP, Subramanya U. Protection of a network by complete secure domination. Journal of Algebra Combinatorics Discrete Structures and Applications. 9(1):47-55.